Domination in regular graphs

A two-valued function f defined on the vertices of a graph G = (V, E), f : V → {-1,1}, is a signed dominating function if the sum of its function values over any closed neighborhoods is at least one. That is, for every υ ∈ V, f(N[υ]) ≥ 1, where N[υ] consists of υ and every vertex adjacent to υ. The function f is a majority dominating function if for at least half the vertices υ ∈ V, f(N[υ]) ≥ 1. The weight of a signed (majority) dominating function is f(V) = Σf(υ), over all vertices υ ∈ V. The signed (majority) domination number of a graph G, denoted γ_a (G) (γ_maj (G), respectively), equals the minimum weight of a signed (majority, respectively) dominating function of G. In this paper, we establish an upper bound on γ_a(G) and a lower bound on γ_maj(G) for regular graphs G.